“Recurrence Equations”, An Integrable System

Abstract

We investige discrete equations which return to the initial states after a finite time regardless of the initial values, which we call “recurrence equations”. The recurrence equation exhibits conserved quantities which are expressed by the fundamental symmetric polynomials of solutions, x_n+1, x_n+2,…, x_n+N to the equations of period N. We show that the recurrence equation is integrable by using algebraic entropy and the conserved quantities. We present new recurrence equations of order k (k≥ 2) which have periods k+1 and 2k+2, respectively.